Bounds for codes on pentagon and other cycles

TL;DR

This paper investigates bounds on the capacity of graphs, especially odd cycles like the pentagon, under fractional error constraints, combining Lovász and Delsarte bounds to determine maximum achievable rates.

Contribution

It derives new upper bounds for the capacity of cycle graphs with fractional error constraints and shows when higher rates are achievable using coding bounds.

Findings

For the pentagon, the Lovász rate is optimal when elta er 1 - 1/5

A Gilbert-Varshamov-type bound shows higher rates are possible for elta < 2/5

The study links graph capacity bounds with coding theory for communication under errors.

Abstract

The capacity of a graph is defined as the rate of exponential grow of independent sets in the strong powers of the graph. In strong power, an edge connects two sequences if at each position letters are equal or adjacent. We consider a variation of the problem where edges in the power graphs are removed among sequences which differ in more than a fraction $\delta$ of coordinates. For odd cycles, we derive an upper bound on the corresponding rate which combines Lov\'asz' bound on the capacity with Delsarte's linear programming bounds on the minimum distance of codes in Hamming spaces. For the pentagon, this shows that for $\delta \ge {1-{1\over\sqrt{5}}}$ the Lov\'asz rate is the best possible, while we prove by a Gilbert-Varshamov-type bound that a higher rate is achievable for $\delta < {2\over 5}$. Communication interpretation of this question is the problem of sending quinary symbols subject to $\pm 1\mod 5$ disturbance. The maximal communication rate subject to the zero undetected-error equals capacity of a pentagon. The question addressed here is how much this rate can be increased if only a fraction $\delta$ of symbols is allowed to be disturbed